Absence of the non-percolating phase for percolation on the non-planar Hanoi network

TL;DR

This paper demonstrates that percolation on the non-planar Hanoi network lacks a non-percolating phase for any positive bond probability, indicating the system is either critical or percolating, supported by analytical bounds and Monte Carlo simulations.

Contribution

It establishes the absence of a non-percolating phase in the non-planar Hanoi network through fractal exponent bounds and simulation evidence.

Findings

No non-percolating phase for p>0

System exhibits either critical or percolating phase

Monte Carlo simulations support the conjecture

Abstract

We investigate bond percolation on the non-planar Hanoi network (HN-NP), which was studied in [Boettcher et al. Phys. Rev. E 80 (2009) 041115]. We calculate the fractal exponent of a subgraph of the HN-NP, which gives a lower bound for the fractal exponent of the original graph. This lower bound leads to the conclusion that the original system does not have a non-percolating phase, where only finite size clusters exist, for p>0, or equivalently, that the system exhibits either the critical phase, where infinitely many infinite clusters exist, or the percolating phase, where a unique giant component exists. Monte Carlo simulations support our conjecture.